A theorem on infinite dimensional group actions

作者：Fejoz Jacques^*; Garay Mauricio

来源：Comptes Rendus Mathematique, 2010, 348(7-8): 427-430.

DOI：10.1016/j.crma.2010.01.024

In this Note, we give an infinitesimal criterion, in an analytic setting, for a vector space to be locally homogeneous under some group action. In the C(infinity) case, such a result was obtained by Sergeraert in his thesis, using an inverse function theorem in Frechet spaces (Sergeraert (1972) [4, Theorem 4.2.5 and Corollary 4.2.6]) (see also Moser (1966) [3], Zehnder (1976) [5]).
Our approach differs from that of Sergeraert because we directly use the underlying group structure. By following the iterative method used by Kolmogorov and Arnold in the proof of the invariant tori theorem (Kolmogorov (1954) [2], Arnold (1963) [1]), it provides an answer to the heuristic question asked by Zehnder (1976) [5. Chap. 5].

• 出版日期2010-4